// Requires CoherentSubgroups.mgm
// In the introduction we make the following claim: every join-coherent
// permutation group of degree at most 11 is either a cyclic group acting
// regularly, a symmetric group, one of the groups described in Theorem 5,
// or an imprimitive wreath product of join-coherent groups of smaller
// degree as in Theorem 2(b). This includes the centralizers in Theorem 3.
// This is easily seen directly for degree <= 4. For higher degrees it
// is useful to use Magma to identify centralizers: these are always
// join-coherent, and evidently they are imprimitive wreath products of
// cyclic and symmetric groups of smaller degree. This leaves a small
// number of other wreath products, and further groups to be identified.
// Degree 5:
// * Dihedral group order 10 (second case Theorem 5)
// * Frobenius group order 20 (also second case Theorem 5)
// * Cyclic group of order 5 acting regularly
> Gs5, Cs5 := JoinCoherentSubgroups(5);
> Gs5;
[
Permutation group acting on a set of cardinality 5
Order = 10 = 2 * 5
(2, 5)(3, 4)
(1, 4, 5, 2, 3),
Permutation group acting on a set of cardinality 5
Order = 20 = 2^2 * 5
(1, 4, 3, 5)
(1, 3)(4, 5)
(1, 4, 2, 5, 3)
]
> Cs5;
[ [*
Permutation group acting on a set of cardinality 5
Order = 5
(1, 4, 3, 2, 5),
(1, 4, 3, 2, 5)
*] ]
// Degree 6:
// * Wreath products C_2 wr C_3 and S_3 wr S_2
// * Centralizers of elements of cycle types (6), (3^2), (2^3)
> Gs6, Cs6 := JoinCoherentSubgroups(6);
> Gs6;
[
Permutation group acting on a set of cardinality 6
Order = 24 = 2^3 * 3
(1, 6, 4)(2, 5, 3)
(1, 3)(2, 6)
(1, 3)(4, 5)
(1, 3)(2, 6)(4, 5),
Permutation group acting on a set of cardinality 6
Order = 72 = 2^3 * 3^2
(1, 4)(2, 6)(3, 5)
(5, 6)
(1, 6)(2, 4)
(1, 6, 5)(2, 4, 3)
(1, 6, 5)(2, 3, 4)
]
> Cs6;
[ [*
Permutation group acting on a set of cardinality 6
Order = 6 = 2 * 3
(1, 5, 4)(2, 6, 3)
(1, 2)(3, 4)(5, 6),
(1, 6, 4, 2, 5, 3)
*], [*
Permutation group acting on a set of cardinality 6
Order = 18 = 2 * 3^2
(1, 3)(2, 5)(4, 6)
(1, 5, 4)(2, 3, 6)
(1, 4, 5)(2, 3, 6),
(1, 4, 5)(2, 3, 6)
*], [*
Permutation group acting on a set of cardinality 6
Order = 48 = 2^4 * 3
(1, 5)(2
, 3)(4, 6)
(1, 3, 6)(2, 5, 4)
(3, 5)(4, 6)
(1, 2)(3, 5)
(1, 2)(3, 5)(4, 6),
(1, 2)(3, 5)(4, 6)
*] ]
// Degree 7:
// * All subgroups of Frobenius group of order 42 (second case Theorem 5)
> Gs7, Cs7 := JoinCoherentSubgroups(7);
> Gs7;
[
Permutation group acting on a set of cardinality 7
Order = 14 = 2 * 7
(2, 4)(3, 6)(5, 7)
(1, 3, 5, 4, 2, 7, 6),
Permutation group acting on a set of cardinality 7
Order = 21 = 3 * 7
(1, 2, 4)(3, 7, 5)
(1, 2, 7, 4, 3, 5, 6),
Permutation group acting on a set of cardinality 7
Order = 42 = 2 * 3 * 7
(2, 3, 5)(4, 6, 7)
(2, 4)(3, 6)(5, 7)
(1, 3, 5, 4, 2, 7, 6)
]
> Cs7;
[ [*
Permutation group acting on a set of cardinality 7
Order = 7
(1, 2, 7, 3, 6, 4, 5),
(1, 2, 7, 3, 6, 4, 5)
*] ]
// Degree 8:
// * Gamma(2^3) from the first case in Theorem 5 (see Proposition 10.1)
// * Wreath products C_2 wr C_4, C_2 wr C_2 wr C_2 and S_4 wr S_2
// * Centralizers of elements of cycle type (8), (4^2), (2^4)
> Gs8, Cs8 := JoinCoherentSubgroups(8);
> Gs8;
[
Permutation group acting on a set of cardinality 8
Order = 16 = 2^4
(1, 2, 3, 8, 5, 4, 6, 7)
(1, 6, 5, 3)(2, 8, 4, 7)
(2, 4)(7, 8)
(1, 5)(3, 6),
Permutation group acting on a set of cardinality 8
Order = 64 = 2^6
(7, 8)
(1, 2, 3, 8, 5, 4, 6, 7)
(3, 6)(7, 8)
(2, 4)(7, 8)
(1, 6)(2, 8)(3, 5)(4, 7)
(1, 5)(3, 6),
Permutation group acting on a set of cardinality 8
Order = 128 = 2^7
(7, 8)
(1, 2, 5, 4)(3, 8)(6, 7)
(3, 6)(7, 8)
(2, 4)(7, 8)
(2, 8)(4, 7)
(1, 5)(3, 6)
(1, 6)(3, 5),
Permutation group acting on a set of cardinality 8
Order = 1152 = 2^7 * 3^2
(7, 8)
(1, 2, 5, 4)(3, 8)(6, 7)
(1, 3)(7, 8)
(4, 8, 7)
(1, 6, 3)
(2, 4)(7, 8)
(2, 8)(4, 7)
(1, 5)(3, 6)
(1, 6)(3, 5)
]
// Checking assertions above:
> S8 := SymmetricGroup(8);
> Gs8[2] eq sub;
> Gs8[3] eq sub;
> Gs8[4] eq sub;
> Cs8;
[ [*
Permutation group acting on a set of cardinality 8
Order = 8 = 2^3
(1, 2, 3, 6, 5, 7, 4, 8)
(1, 4, 5, 3)(2, 8, 7, 6)
(1, 5)(2, 7)(3, 4)(6, 8),
(1, 2, 3, 6, 5, 7, 4, 8)
*], [*
Permutation group acting on a set of cardinality 8
Order = 32 = 2^5
(2, 6, 7, 8)
(1, 2, 3, 6, 5, 7, 4, 8)
(1, 4, 5, 3)(2, 6, 7, 8)
(2, 7)(6, 8)
(1, 5)(3, 4),
(1, 3, 5, 4)(2, 6, 7, 8)
*], [*
Permutation group acting on a set of cardinality 8
Order = 384 = 2^7 * 3
(3, 8)(4, 6)
(2, 4, 6)(3, 8, 7)
(1, 3)(2, 6)(4, 5)(7, 8)
(6, 8)
(1, 7)(2, 5)(3, 8)(4, 6)
(2, 7)(6, 8)
(3, 4)(6, 8)
(1, 5)(2, 7)(3, 4)(6, 8),
(1, 5)(2, 7)(3, 4)(6, 8)
*] ]
// Degree 9:
// * Gamma(3^2) from first case in Theorem 5 (see Proposition 10.1)
// * Wreath products C_3 wr C_3, S_3 wr C_3, S_3 wr S_3
// * Centralizers of elements of cycle type (9) and (3^3)
> Gs9, Cs9 := JoinCoherentSubgroups(9);
> Gs9;
[
Permutation group acting on a set of cardinality 9
Order = 27 = 3^3
(1, 7, 4, 3, 8, 5, 2, 9, 6)
(1, 3, 2)(7, 9, 8)
(4, 5, 6)(7, 9, 8),
Permutation group acting on a set of cardinality 9
Order = 81 = 3^4
(1, 8, 5)(2, 7, 4)(3, 9, 6)
(1, 3, 2)(7, 8, 9)
(4, 5, 6)(7, 8, 9)
(7, 8, 9),
Permutation group acting on a set of cardinality 9
Order = 648 = 2^3 * 3^4
(8, 9)
(1, 8, 5)(2, 7, 4)(3, 9, 6)
(5, 6)(8, 9)
(2, 3)(4, 5, 6)(8, 9)
(1, 3, 2)(7, 8, 9)
(4, 5, 6)(7, 8, 9)
(7, 8, 9),
Permutation group acting on a set of cardinality 9
Order = 1296 = 2^4 * 3^4
(1, 2, 3)
(4, 5, 6)
(1, 5, 2, 4)(3, 6)
(1, 4, 7)(2, 5, 8)(3, 6, 9)
(1, 7, 2, 8)(3, 9)(4, 5)
]
// Checking assertions above:
> S9 := SymmetricGroup(9);
> Gs9[2] eq sub;
> Gs9[3] eq sub;
> Gs9[4] eq sub;
> Cs9;
[ [*
Permutation group acting on a set of cardinality 9
Order = 9 = 3^2
(1, 7, 5, 3, 8, 6, 2, 9, 4)
(1, 3, 2)(4, 5, 6)(7, 8, 9),
(1, 7, 5, 3, 8, 6, 2, 9, 4)
*], [*
Permutation group acting on a set of cardinality 9
Order = 162 = 2 * 3^4
(4, 7, 6, 8, 5, 9)
(1, 7, 4, 2, 8, 6, 3, 9, 5)
(4, 5, 6)(7, 8, 9)
(1, 3, 2)(7, 8, 9)
(7, 8, 9),
(1, 3, 2)(4, 5, 6)(7, 9, 8)
*] ]
// Degree 10:
// * Wreath products D_5 wr C_2, C_2 wr C_5, C_2 wr D_5, F_20 wr C_2, C_2 wr F_20
// * Centralizers of elements of cycle type (10), (5^2) and (2^5)
// where D_5 denotes the dihedral group of order 10 and F_20 the Frobenius
// group of order 20 (see degree 5 classification)
> Gs10, Cs10 := JoinCoherentSubgroups(10);
> Gs10;
[
Permutation group acting on a set of cardinality 10
Order = 200 = 2^3 * 5^2
(6, 10)(7, 8)
(1, 6)(2, 9)(3, 7)(4, 8)(5, 10)
(1, 5)(3, 4)(6, 10)(7, 8)
(1, 5, 4, 2, 3)
(6, 8, 7, 10, 9),
Permutation group acting on a set of cardinality 10
Order = 160 = 2^5 * 5
(1, 9, 3, 6, 8)(2, 10, 4, 5, 7)
(1, 2)
(3, 4)
(5, 6)
(7, 8)
(9, 10),
Permutation group acting on a set of cardinality 10
Order = 320 = 2^6 * 5
(1, 8)(2, 7)(5, 10)(6, 9)
(1, 9, 3, 6, 8)(2, 10, 4, 5, 7)
(1, 2)
(3, 4)
(5, 6)
(7, 8)
(9, 10),
Permutation group acting on a set of cardinality 10
Order = 800 = 2^5 * 5^2
(7, 8, 9, 10)
(1, 7, 5, 10, 3, 9, 4, 8)(2, 6)
(7, 9)(8, 10)
(1, 5, 3, 4)(7, 8, 9, 10)
(1, 3)(4, 5)(7, 9)(8, 10)
(6, 7, 8, 10, 9)
(1, 3, 4, 2, 5),
Permutation group acting on a set of cardinality 10
Order = 640 = 2^7 * 5
(1, 2)(3, 6, 9, 8, 4, 5, 10, 7)
(1, 2)(3, 9)(4, 10)(5, 7, 6, 8)
(1, 7, 9, 3, 6)(2, 8, 10, 4, 5)
(1, 2)
(3, 4)
(5, 6)
(7, 8)
(9, 10)
]
// Checking assertions above:
> S10 := SymmetricGroup(10);
> Gs10[1] eq sub;
> Gs10[2] eq sub;
> Gs10[3] eq sub;
> Gs10[4] eq sub;
> Gs10[5] eq sub;
> Cs10;
[ [*
Permutation group acting on a set of cardinality 10
Order = 10 = 2 * 5
(1, 9, 3, 6, 8)(2, 10, 4, 5, 7)
(1, 2)(3, 4)(5, 6)(7, 8)(9, 10),
(1, 10, 3, 5, 8, 2, 9, 4, 6, 7)
*], [*
Permutation group acting on a set of cardinality 10
Order = 50 = 2 * 5^2
(1, 6)(2, 7)(3, 8)(4, 10)(5, 9)
(6, 7, 10, 8, 9)
(1, 4, 5, 2, 3),
(1, 2, 4, 3, 5)(6, 7, 10, 8, 9)
*], [*
Permutation group acting on a set of cardinality 10
Order = 3840 = 2^8 * 3 * 5
(1, 3, 5, 7, 9)(2, 4, 6, 8, 10)
(1, 3)(2, 4)
(1, 2),
(1, 2)(3, 4)(5, 6)(7, 8)(9, 10)
*] ]
// Degree 11:
// * All subgroups of Frobenius group of order 110 (second case Theorem 5)
// * Cyclic group of order 11 acting regularly
> Gs11, Cs11 := JoinCoherentSubgroups(11);
> Gs11;
[
Permutation group acting on a set of cardinality 11
Order = 22 = 2 * 11
(2, 11)(3, 10)(4, 9)(5, 8)(6, 7)
(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11),
Permutation group acting on a set of cardinality 11
Order = 55 = 5 * 11
(2, 5, 6, 10, 4)(3, 9, 11, 8, 7)
(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11),
Permutation group acting on a set of cardinality 11
Order = 110 = 2 * 5 * 11
(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11)
(1, 2, 4, 8, 5, 10, 9, 7, 3, 6)
]
> Cs11;
[ [*
Permutation group acting on a set of cardinality 11
Order = 11
(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11),
(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11)
*] ]
// Degree 12 example from introduction
S12 := SymmetricGroup(12);
G := sub;
testG1 := TestSubgroup(Join,G);
testG2 := G subset sub;